How critical values work
A critical value marks the boundary of the rejection region in a hypothesis test. If the test statistic falls beyond the critical value, the null hypothesis is rejected. For a two-tailed test at α = 0.05, the z critical values are ±1.96, meaning the most extreme 5% of the distribution is split between both tails.
For t-tests and chi-square tests, critical values depend on the degrees of freedom. As degrees of freedom increase, t critical values approach z critical values. Chi-square critical values are always right-tailed because the chi-square distribution is non-negative.
zₐ = Φ⁻¹(1 − α)
Right-tailed z critical value from the inverse standard normal CDF.
z_{α/2} = Φ⁻¹(1 − α/2)
Two-tailed z critical value.